UGrid: An Efficient-And-Rigorous Neural Multigrid Solver for Linear PDEs
Numerical solvers of Partial Differential Equations (PDEs) are of fundamental significance to science and engineering. To date, the historical reliance on legacy techniques has circumscribed possible integration of big data knowledge and exhibits sub-optimal efficiency for certain PDE formulations, while data-driven neural methods typically lack mathematical guarantee of convergence and correctness. This paper articulates a mathematically rigorous neural solver for linear PDEs. The proposed UGrid solver, built upon the principled integration of U-Net and MultiGrid, manifests a mathematically rigorous proof of both convergence and correctness, and showcases high numerical accuracy, as well as strong generalization power to various input geometry/values and multiple PDE formulations. In addition, we devise a new residual loss metric, which enables unsupervised training and affords more stability and a larger solution space over the legacy losses.
Code (1)
Methods 이 논문이 사용한 방법론
Similar Papers 제목 키워드 기반
Learning to Optimize Multigrid PDE Solvers
Constructing fast numerical solvers for partial differential equations (PDEs) is crucial for many scientific disciplines. A leading technique for solving large-scale PDEs is using multigrid methods. At the core of a mult…
NSPOD: Accelerating Krylov solvers via DeepONet-learned POD subspaces
The convergence of Krylov-based linear iterative solvers applied to parametric partial differential equations (PDEs) is often highly sensitive to the domain, its discretization, the location/values of the applied Dirichl…
Automating the Design of Multigrid Methods with Evolutionary Program Synthesis
Many of the most fundamental laws of nature can be formulated as partial differential equations (PDEs). Understanding these equations is, therefore, of exceptional importance for many branches of modern science and engin…
Code GenerationProgram SynthesisMechanistic PDE Networks for Discovery of Governing Equations
We present Mechanistic PDE Networks -- a model for discovery of governing partial differential equations from data. Mechanistic PDE Networks represent spatiotemporal data as space-time dependent linear partial differenti…
GPUGamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexit…